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By H. F. Blichfeldt

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Example text

Neither zero nor all of V ) G-invariant subspace of V . It follows that a nonzero ﬁnite-dimensional representation V is irreducible if and only if V = spanC {gv | g ∈ G} for each nonzero v ∈ V , since this property is equivalent to excluding proper Ginvariant subspaces. 1 and so each is irreducible. For more general representations, this approach is often impossible to carry out. In those cases, other tools are needed. One important tool is based on the next result. 12 (Schur’s Lemma). Let V and W be ﬁnite-dimensional representations of a Lie group G.

This one-dimensional representation is called the trivial representation. More generally, the action of G on a vector space is called trivial if each g ∈ G acts as the identity operator. 1 Standard Representations Let G be G L(n, F), S L(n, F), U (n), SU (n), O(n), or S O(n). The standard representation of G is the representation on Cn where π(g) is given by matrix multiplication on the left by the matrix g ∈ G. It is clear that this deﬁnes a representation. 2 SU(2) This example illustrates a general strategy for constructing new representations.

Since ∗top (G/H )eH is one-dimensional, lh∗ ωeH = c(h)ωeH for h ∈ H and some c(h) ∈ R\{0}. The equality lhh = lh ◦ lh shows that c : H → R\{0} is a homomorphism. The compactness of G shows that c(h) ∈ {±1}. If H is connected, the image of H under c must be connected and so c(h) = 1, which shows that ωG/H exists. Suppose that ωG/H exists. Since dh is invariant, the function g → H f (gh) dh may be viewed as a function on G/H . Working with characteristic functions, the assignment f → G/H H f (gh) dh d(g H ) deﬁnes a normalized left invariant Borel measure on G.